Question

Explain polynomial and factorisation.

Answer 1

A polynomial is an expression that can have variables,constants, that can be combined using addition, subtraction, multiplication and division.

For example: 2/(x+2) is not a polynomial but it is an expression.
Polynomials can be classified into different types of polynomials based upon
1. Number of variables in Polynomial
For example: $x^3+x^2$ is a polynomial in one variable
xy is a polynomial in two variables.
2. Number of terms present in polynomial
For example: $x^3$ is a monomial as it has only one term
$x^2+x^3$ is binomial as it has two terms.
Factorization of Polynomial
Factorization of Polynomial is same as factorization of numbers.
For example 10 can be written as $1\ast 2\ast 5$. In similar way we can factorize a polynomial.
For example: $p\left(x\right)=x^2+5x+6$ can be factorized as (x+2)(x+3). For polynomial $x^2+5x+6,(x+2)$ and (x+3) are the factors.
Here you will find that if you substitute x= -2 then you will get p(-2) = 0. Similarly if you substitute x= -3 you will get p(-3)=0.
We got p(-2)=0, p(-3)=0 as (x+2) and (x+3) are factors of $x^2+5x+6.$
Here comes the factor theorem that states that If p(x) is a polynomial, then
i) (x-a) is a factor of p(x), if p(a)= 0, and
ii) p(a)=0, if (x-a) is a factor of p(x).

So if it is given to you in a question that for a polynomial p(x), p(5)=0 then you can say that (x-5) is a factor of polynomial p(x).